Second-order Expansion Research Articles

Depression recognition using high-order generalized multilayer brain functional network fused with EEG multi-domain information

Major Depressive Disorder (MDD) is a serious and highly heterogeneous psychological disorder. According to the network hypothesis, depression originates from abnormal neural network information processing, typically resulting in aberrant changes in the topological structure of the brain’s functional network. Recent evidence further reveals that depression involves dynamic changes related to both within- and cross-frequency coupling. Therefore, we utilize second-order tensor expansion to integrate frequency- and time-varying multilayer brain functional networks based on node sharing, thus propose a generalized multilayer brain functional network (GMBFN) incorporating multi-domain information. Concurrently, we derive global and local topological properties from both the frequency and temporal domains to characterize the novel network structure. To uncover more reliable biomarkers and explore various coupling features that can assess the interaction between signals from different perspectives, we conduct research in two datasets employing four sets of within- and cross-frequency coupling. Leveraging the novel multi-domain high-order GMBFNs, abnormalities of information integration abilities in patients with MDD are observed, particularly in the theta-band and overall temporal-domain. Through the fusion of topological properties across both domains with multiple classifiers, the alpha-band can serve as a potential biomarker for depression identification. More importantly, the combination of global topological properties from both domains, on average, enhances the classification performance for identifying patients with MDD by 5.18% compared to using just one domain. This study presents a systematic framework for comprehending the aberrant mechanisms of MDD from multiple perspectives, offering significant value for clinical applications aimed at assisting in depression diagnosis and intervention.

Real-time phase-resolved ocean wave prediction in directional wave fields: Second-order Lagrangian wave models

The Improved Choppy Wave Model (ICWM) was established to achieve a second-order Lagrangian expansion of surface waves. The second-order Lagrangian nonlinear interaction terms were found to be negligible and were therefore discarded. However, these interactions in directional wave fields remained unexplored. ICWM was successfully used, especially for floating offshore wind turbines, but it was noted that its accuracy in predicting directional sea states needed improvement. Hence, we formulated the Complementary ICWM (CICWM) with the nonlinear terms for free surface elevation in directional waves. Detailed formulations for data assimilation and wave propagation were provided, enabling real-time wave forecasting for directional seas using a simplified assimilation method. Comparing the model performances against tank-scale experiments showed that CICWM enhances the surface elevation description in directional seas. Ultimately, it was confirmed that the second-order Lagrangian model can reduce the prediction error of the linear wave model by 90% in directional seas for the experimental setups and sea states investigated in this study.

A reliability analysis method based on the intersection area division of hypersphere and paraboloid

In terms of the approximate failure probability of the limit state function (LSF), First Order Reliability Method (FORM) directly solves the failure probability through the most probable point (MPP), but it can only achieve high accuracy in the LSF with low linearity and non-linearity. Second Order Reliability Method (SORM) performs second-order expansion on the basis of MPP to improve reliability accuracy. However, the approximation may be wrong, and even has a large error in high-dimensional nonlinear problems. In order to get more precise approximation of SORM, this paper proposes an area partition method based on the Intersection Area Division Method (IADM) of Hypersphere and Paraboloid. IADM constructs a hypersphere and an approximate parabolic function of the LSF at MPP. The intersection area of the hypersphere and the approximate paraboloid is defined as the failure region from the geometric of the approximate paraboloid function, and the area of the failure region is calculated by the spherical integral. Finally, the approximation of the failure probability is determined by integrating the ratio of the area to the area of the hypersphere. Several examples proved that IADM is more stable compared with SORM which avoids the generation of wrong solutions and improves the accuracy.

New Functional Orbital-free Within DFT for Metallic Systems

I present the continuation of a study on Laplacian Level Kinetic Energy (KE) functionals applied to metallic nanosystems. The development of novel Kinetic Energy functionals is an important topic in density functional theory (DFT). The nanoparticles are patterned using gelatin spheres of different sizes, background density and number of electrons. To reproduce the correct kinetic and potential energy density of the various nanoparticles, the use of semilocal descriptors is necessary. Need an explicit density functional expression for the kinetic energy of electrons, including the first e second functional derivative, i.e. the kinetic potential and the kinetic kernel, respectively. The exact explicit form of the non interacting kinetic energy, as a functional of the electron density, is known only for the homogeneous electron gas (HEG), i.e., the Thomas-Fermi (TF) local functional and for 1 and 2 electron systems, i.e., the von Weizsacker (VW) functional. In between these two extreme cases, different semilocal or non local approximations were developed in recent years. Most semilocal KE functionals are based on modifications of the second-order gradient expansion (GE2) or fourth-order gradient expansion (GE4). I find that the Laplacian contribute is fundamental for the description of the energy and the potential of nanoparticles. I propose a new LAP2 semilocal functional which, better than the previous ones, allows us to obtain fewer errors both of energy and potential. More details of the previous calculations can be found in my 2 previous works which will be cited in the text.

Statistical distribution of static stress resolved onto geometrically-rough faults

The in-situ stress state within fault zones is technically challenging to characterize, requiring the use of indirect methods to estimate. Most work to date has focused on understanding average properties of resolved stress on faults, but fault non-planarity should induce spatial variations in resolved static stress on a single fault. Assuming a particular stochastic model for fault geometry (band-limited fractal) gives an approximate analytic solution for the probability density function (PDF) on fault stress that depends on the mean fault orientation, mean stress ratio, and roughness level. The mean stress is shown to be equal to the planar fault value, while deviations are described by substituting a second-order polynomial expansion of the stress ratio into the inverse distribution on fault slope. The result is an analytical expression for the PDF of shear-to-normal stress ratio on 2-D rough faults in a uniform background stress field. Two end-member distributions exist, one approximately Gaussian when all points on the fault are well away from failure, and one reverse exponential, which occurs when the mean stress ratio approaches the peak. For the range of roughness values expected to apply to crustal faults, stress deviations due to geometry can reach nearly 100% of the background stress level. Consideration of such a distribution of stress on faults suggests that geometric roughness and the resulting stress deviations may play a key role in controlling earthquake behavior.

Nonlinear electrodynamics for the vacuum of Dirac materials: Photon magnetic properties and radiation pressures

We investigate the magnetic properties of photons propagating through Dirac materials in a magnetic field, considering both vacuum and medium contributions. Photon propagation properties are obtained through a second-order expansion of nonlinear Euler-Heisenberg electrodynamics at finite density and temperature considering Dirac material parameters (Dirac fine structure constant, band gap, and Fermi velocity). Total magnetization (including electron and photon contributions) and photon-effective magnetic moment are computed. Observables such as photon energy density, radiation pressure, and Poynting vector are obtained by an average of components of the energy-momentum tensor. All quantities are expressed in terms of Lagrangian derivatives. Those related to the vacuum are valid for any value of the external magnetic field, and both the weak- and strong-field limits are recovered. We discuss some ideas of experiments that may contribute to testing in Dirac materials the phenomenology of the strong magnetic field in the quantum electrodynamic (QED) vacuum and how nonlinear corrections on the magnetization, radiation pressure, and birefringence are amplified up to 103 times QED corrections. Published by the American Physical Society 2024

Conforming Capacitive Load Cells for Conical Pick Cutters.

In underground coal mining, machine operators put themselves at risk when getting close to the machine or cutting face to observe the process. To improve the safety and efficiency of machine operators, a cutting force sensor is proposed. A linear cutting machine is used to cut two separate coal samples cast in concrete with conical pick cutters to simulate mining with a continuous miner. Linear and neural network regression models are fit using 100 random 70:30 test/train splits. The normal force exceeds 60 kN during the rock-cutting tests, and it is averaged using a low pass filter with a 10 Hertz cutoff frequency. The sensor uses measurements of the resonant frequency of capacitive cells in a steel case to determine cutting forces. When used in the rock-cutting experiments, the sensor conforms to the tooling and the stiffness and sensitivity are increased compared to the initial configuration. The sensor is able to track the normal force on the conical picks with a mean absolute error less than 6 kN and an R2 score greater than 0.60 using linear regression. A small neural network with a second-order polynomial expansion is able to improve this to a mean absolute error of less than 4 kN and an R2 score of around 0.80. Filtering measurements before regression fitting is explored. This type of sensor could allow operators to assess tool wear and material type using objective force measurements while maintaining a greater distance from the cutting interface.

Note on the bispectrum and one-loop corrections in single-field inflation with primordial black hole formation

Primordial black holes can be formed from the collapse of large-amplitude perturbation on small scales in the early Universe. Such an enhanced spectrum can be realized by introducing a flat region in the potential of single-field inflation, which makes the inflaton go into a temporary ultraslow-roll period. In this paper, we calculate the bispectrum of curvature perturbation in such a scenario. We explicitly confirm that bispectrum satisfies Maldacena’s theorem. At the end of the ultraslow-roll period, the bispectrum is generated by bulk interaction and field redefinition. At the end of inflation, bispectrum is generated only by bulk interaction. We also calculate the one-loop correction to the power spectrum from the bispectrum, called the source method. We find it consistent with the calculation of the one-loop correction from the second-order expansion of in-in perturbation theory. Published by the American Physical Society 2024

Theory of 2D electronic spectroscopy of water soluble chlorophyll-binding protein (WSCP): Signatures of Chl b derivate.

We investigate how electronic excitations and subsequent dissipative dynamics in the water soluble chlorophyll-binding protein (WSCP) are connected to features in two-dimensional (2D) electronic spectra, thereby comparing results from our theoretical approach with experimental data from the literature. Our calculations rely on third-order response functions, which we derived from a second-order cumulant expansion of the dissipative dynamics involving the partial ordering prescription, assuming a fast vibrational relaxation in the potential energy surfaces of excitons. Depending on whether the WSCP complex containing a tetrameric arrangement of pigments composed of two dimers with weak excitonic coupling between them binds the chlorophyll variant Chl a or Chl b, the resulting linear absorption and circular dichroism spectra and particularly the 2D spectra exhibit substantial differences in line shapes. These differences between Chl a WSCP and Chl b WSCP cannot be explained by the slightly modified excitonic couplings within the two variants. In the case of Chl a WSCP, the assumption of equivalent dimer subunits facilitates a reproduction of substantial features from the experiment by the calculations. In contrast, for Chl b WSCP, we have to assume that the sample, in addition to Chl b dimers, contains a small but distinct fraction of chemically modified Chl b pigments. The existence of such Chl b derivates has been proposed by Pieper et al. [J. Phys. Chem. B 115, 4042 (2011)] based on low-temperature absorption and hole-burning spectroscopy. Here, we provide independent evidence.

Theory and practice of second-order expansions for moments of 100ρ% accelerated sequential stopping times in parametric and nonparametric estimation with arbitrary fractional ρ

Sampling strategies with fractional acceleration 0 < ρ < 1 achieve substantial operational savings compared with purely sequential counterparts. But, acceleration customarily yielded second-order (s.o.) lower and upper bounds for requisite characteristics when ρ − 1 was not an integer. First time in the literature, we have recently designed acceleration with asymptotic s.o. expansions in normal mean problems with ρ − 1 arbitrary. In this article, a general unified theory is now developed leading to asymptotic s.o. expansions for customarily studied characteristics with arbitrary ρ − 1 . That is, the previously known s.o. lower/upper bounds can be replaced with appropriate s.o. expansions in a variety of inference problems with prescribed accuracy. We emphasize the theoretical foundation and ensuing analyses with a series of interesting inference problems from (a) a number of non-normal distributions as well as (b) one- and two-sample distribution-free scenarios. These prove a desirable breadth of coverage under our proposed big tent.

Hybrid bias and displacement emulators for field-level modelling of galaxy clustering in real and redshift space

ABSTRACT Recently, hybrid bias expansions have emerged as a powerful approach to modelling the way in which galaxies are distributed in the Universe. Similarly, field-level emulators have recently become possible, thanks to advances in machine learning and N-body simulations. In this paper, we explore whether both techniques can be combined to provide a field-level model for the clustering of galaxies in real and redshift space. Specifically, here we will demonstrate that field-level emulators are able to accurately predict all the operators of a second-order hybrid bias expansion. The precision achieved in real and redshift space is similar to that obtained for the non-linear matter power spectrum. This translates to roughly 1–2 per cent precision for the power spectrum of a BOSS (Baryon Oscillation Spectroscopic Survey) and a Euclid-like galaxy sample up to $k\sim 0.6\ h\, {\rm Mpc}^{-1}$. Remarkably, this combined approach also delivers precise predictions for field-level galaxy statistics. Despite all these promising results, we detect several areas where further improvements are required. Therefore, this work serves as a road map for the developments required for a more complete exploitation of upcoming large-scale structure surveys.

An instance of negative variance-like ‘components’ in optics by way of applying multivariable calculus to a statistical problem

In the field of optics, the design of a lens often involves analyses of the variability in an optical parameter (such as lens focal length) under manufacturing conditions, which is done in order to identify sources of high variability that may cause the optical parameter to be out of its specification. In this interdisciplinary paper, the author shares his experience in lens design, where analyses of variability are often performed through examining the partial derivatives in a multivariable Taylor series expansion of an optical parameter to second order when interactions are expected. It is shown algebraically that when the second-order expansion is applied to express a sum of squares, negative terms in the sum can occur. This instance of negative-valued “variance-like components” is interesting pedagogically in applied mathematics and statistics, as well as from an engineering standpoint, as it suggests that, based on theory, the variance of a function of random variables may not necessarily be comprised of a simple sum of positive-valued squared deviations comprising its total variance.

Calculation of shift current tensors in two-dimensional transition metal dichalcogenides

We investigate the shift current induced by optical excitation in two-dimensional transition metal dichalcogenides. Our approach is based on a second-order perturbation expansion of the semiconductor Bloch equations which are formulated in the basis of a three-band tight-binding model. From that, shift current tensors have been calculated for several different materials and their dependence on photon energy has been investigated and discussed.

Convergence criteria for single-step free-energy calculations: the relation between the Π bias measure and the sample variance.

Free energy calculations play a crucial role in simulating chemical processes, enzymatic reactions, and drug design. However, assessing the reliability and convergence of these calculations remains a challenge. This study focuses on single-step free-energy calculations using thermodynamic perturbation. It explores how the sample distributions influence the estimated results and evaluates the reliability of various convergence criteria, including Kofke's bias measure Π and the standard deviation of the energy difference ΔU, σ ΔU . The findings reveal that for Gaussian distributions, there is a straightforward relationship between Π and σ ΔU , free energies can be accurately approximated using a second-order cumulant expansion, and reliable results are attainable for σ ΔU up to 25 kcal mol-1. However, interpreting non-Gaussian distributions is more complex. If the distribution is skewed towards more positive values than a Gaussian, converging the free energy becomes easier, rendering standard convergence criteria overly stringent. Conversely, distributions that are skewed towards more negative values than a Gaussian present greater challenges in achieving convergence, making standard criteria unreliable. We propose a practical approach to assess the convergence of estimated free energies.

Revisiting physical mechanism of longitudinal photonic spin splitting and Goos-Hänchen shift

The intrinsic connection between the transverse photonic spin Hall effect (PSHE) and the Imbert–Fedorov shift has been well characterized. However, physical insights into the longitudinal photonic spin splitting associated with the Goos-Hänchen (GH) shift remain elusive. This paper aims to expand the theory of the PSHE generation mechanism from the transverse to the longitudinal case by examining the reflection of each spin component from an arbitrarily linearly polarized incident Gaussian beam on the air-dielectric interface. Unlike the transverse case, both spin-maintained and spin-flipped modes exhibit non-zero longitudinal displacements, with the latter being affected by the second-order expansion term of the Fresnel reflection coefficient with respect to the in-plane wave-vector component. Meanwhile, the polarization angle plays a crucial role in determining the longitudinal PSHE since each reflected total spin component is a coherent superposition of these two corresponding modes. Remarkably, the imaginary part of the relative permittivity of the dielectric significantly affects the symmetry of the longitudinal PSHE. Furthermore, the GH shift results from a superposition of individual spin states’ longitudinal displacements, taking into account their energy weights. By incorporating the corresponding extrinsic orbital angular momentum, we explore the generation mechanism of the symmetric/asymmetric longitudinal PSHE. The unified physical framework elucidating the longitudinal photonic spin splitting and GH shift provides a comprehensive understanding of the fundamental origin of the PSHE and beam shifts, paving the way for potential applications in spin-controlled nanophotonics.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Stochastic stability and the moment Lyapunov exponent for a gyro-pendulum system driven by a bounded noise

Abstract. The stochastic stability of a gyro-pendulum system parametrically excited by a real noise is investigated by the moment Lyapunov exponent in the paper. Using the spherical polar and non-singular linear stochastic transformations and combining these with Khasminskii's method, the diffusion process and the eigenvalue problem of the moment Lyapunov exponent are obtained. Then, applying the perturbation method and Fourier cosine series expansion, we derive an infinite-order matrix whose leading eigenvalue is the second-order expansion g2(p) of the moment Lyapunov exponent. Thus, an infinite sequence for g2(p) is constructed, and its convergence is numerically verified. Finally, the influences of the system and noise parameters on stochastic stability are given such that the stochastic stability is strengthened with the increased drift coefficient and the diffusion coefficient has the opposite effect; among the system parameters, only the increase in k and A0 strengthens moment stability.

Reworking the Tao-Mo exchange-correlation functional. II. De-orbitalization.

In Paper I [H. Francisco, A. C. Cancio, and S. B. Trickey, J. Chem. Phys. 159, 214102 (2023)], we gave a regularization of the Tao-Mo exchange functional that removes the order-of-limits problem in the original Tao-Mo form and also eliminates the unphysical behavior introduced by an earlier regularization while essentially preserving compliance with the second-order gradient expansion. The resulting simplified, regularized (sregTM) functional delivers performance on standard molecular and solid state test sets equal to that of the earlier revised, regularized Tao-Mo functional. Here, we address de-orbitalization of that new sregTM into a pure density functional. We summarize the failures of the Mejía-Rodríguez and Trickey de-orbitalization strategy [Phys. Rev. A 96, 052512 (2017)] when used with both versions. We discuss how those failures apparently arise in the so-called z' indicator function and in substitutes for the reduced density Laplacian in the parent functionals. Then, we show that the sregTM functional can be de-orbitalized somewhat well with a rather peculiarly parameterized version of the previously used deorbitalizer. We discuss, briefly, a de-orbitalization that works in the sense of reproducing error patterns but that apparently succeeds by cancelation of major qualitative errors associated with the de-orbitalized indicator functions α and z, hence, is not recommended. We suggest that the same issue underlies the earlier finding of comparatively mediocre performance of the de-orbitalized Tao-Perdew-Staroverov-Scuseri functional. Our work demonstrates that the intricacy of such two-indicator functionals magnifies the errors introduced by the Mejía-Rodríguez and Trickey de-orbitalization approach in ways that are extremely difficult to analyze and correct.

Quasi-classical ground states. I. Linearly coupled Pauli–Fierz Hamiltonians

We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy \mathcal{E}(u,f) of product states of the form u\otimes\Psi_f , where u is a normalized state for the particle and \Psi_f is a coherent state in Fock space for the field, gives the energy of a Klein–Gordon–Schrödinger system. We minimize the functional \mathcal{E}(u,f) on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein–Gordon–Schrödinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.

Compressibility-induced destabilisation of falling liquid films: an integral approach

We revisit the classical 2D problem of a gravity-driven liquid layer down an inclined plate (Kapitza, 1948), relaxing the usual assumption of homogeneous fluid. We set out to answer three major issues. When the fluid density is allowed to vary, (i) how does this feature structurally affect the formulation of a low-dimensional depth-averaged model? (ii) To what extent and (iii) by virtue of which physical mechanism does compressibility participate in the long-wave interfacial instability? To provide the relevant answers, (i) we first make use of a second-order asymptotic expansion in the shallowness parameter to develop a weakly-compressible boundary-layer system: starting from a two-equation momentum-integrated model, an additional barotropic equation of state is required for closure purposes. In this respect, (ii) a temporal linear stability analysis is performed: it is revealed that compressibility plays a destabilising role whose magnitude is enhanced at intermediately tilted configurations, and the more the Reynolds number approaches the critical threshold in the incompressible limit. (iii) We finally interpret the ensuing dispersion relation under the convenient framework of two-wave hierarchy, initiated by Whitham (1974): the primary instability gets promoted by the flow compressibility as it contributes to deceleration of dynamic waves most significantly in the low-inertia regime. Indeed, compressibility locally acts as a further boost to the inertia-based mechanism of Kapitza instability by amplifying flow-rate variations within the liquid film.